Poole Ch.P., Jr. Handbook of Superconductivity

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506 Chapter 10: Thermal Properties

Fig. 10.8.

>, 15

........

I I I I I

0 -- ~ I I ....... !. I

0 50 100 150 200 250 300

Temperature

(K)

Anisotropy ratio

Kab/K c

temperature dependence for crystals of high-T c superconductors: (3,

YBa2Cu306. 7 (Sera

et aL,

1990); 0, YBa2Cu307_ 6 (Hagen

et al.,

1989); T, YBa2Cu307 Cao

et al.,

1991); D, YBa2Cu307_,~ (Efunov and Mezhov-Deglin, 1997); (~, Bi2Sr2CaCuOs_ x

(Efimov and Mezhov-Deglin, 1997); A, Bi2Sr2CaCuO 8 (Crommie and Zettl, 1991); II,

T12Ba2CaCu208 (Cao

et al.,

1991), O, Lal.96Sr0.04CuO 4 (Morelli

et al.,

1990).

quasiparticle. This process may drastically alter the heat flow because of the near

reversal of the group velocity.

Phonon-vortex scattering can be studied only at sufficiently low tempera-

tures where the lattice carries most of the heat. For phonons to scatter, their

wavelength must be comparable to the cross-section of the vortex. Phonon

scattering intensifies with increasing vortex density as

wp(~)- Wp(0)[1 + ~t~(~/Bc2)],

(33)

where a is the average scattering diameter of the vortex, and lp is the phonon mfp.

Again, as the vortices come close together, the electrons start to tunnel between

them and the conductivity approaches the normal state as B ~ Bc2. Typical

behavior of the magnetothermal conductivity in conventional superconductors is

sketched in Fig. 10.9. It is important to note that Eqs. (32) and (33) predict the

same field dependence, and thus the magnetic field, on its own, cannot distinguish

between the effects due to quasiparticles and those due to phonons.

A magnetic field has a strong influence on the heat transport in HTS single

crystals and causes up to 30% reduction in the thermal conductivity (Palstra

et aL

1990); Florentiev

et al.,

1990; Zavaritski

et al.,

1991; Peacor

et aL,

199 lb). The

results show interesting hysteresis effects associated with the flux creep phenom-

enon, and this provides a way to study vortex dynamics via thermal transport

measurements (Richardson

et aL,

1991). Vortices scatter more effectively when

C. Thermal Conductivity 507

Fig. 10.9.

K(H)

(a)

K,,

1 , I

Hcx Hc 2

K(H)

.~H ~H

(b)

I I

Hc I Hc 2

Effect of magnetic field on the thermal conductivity of conventional superconductors. (a)

Behavior of a typical superconducting alloy. (b) Behavior of a "clean" superconductor.

they lie perpendicular to the

CuO 2

planes than when they fit between the planes

(see Fig. 10.10), and dimensionality of superconductivity has a considerable

influence (Inyushkin

et al.,

1994). In marked contrast to conventional super-

conductors, the field dependence of thermal conductivity in HTS is substantially

sublinear, and stretched exponentials of the form

Wo(B ) -- cB exp[-pB1/4],

(34)

with c andp being fitting parameters, provide excellent fits to the data (Fig. 10.11;

Richardson

et al.,

1991). As yet, there is no consensus regarding the nature of this

sublinear field dependence, (Richardson

et al.,

1991; Bougrine

et al.,

1993;

Sergeenkov and Ausloos, 1995). Again, because of a similar functional depen-

dence of the quasiparticle and phonon contributions on the magnetic field

strength, one cannot assess the relative contributions of quasiparticles and

phonons without some a priori information concerning the dominant carrier.

However, the nontraditional transport techniques that make use of the

handedness in scattering of quasiparticles on vortices offer great hope for

distinguishing between the contributions of quasiparticles and phonons. One

such approach is the Righi-Leduc configuration used by Krishana

et al. (1995);

see Fig. 10.12. Although phonons are scattered symmetrically by the vortices,

there is considerable asymmetry (handedness) as the quasiparticles encounter

currents circling around the vortex core. Consequently, the transverse thermal

conductivity Kxy is given entirely by quasiparticles without any phonon back-

ground. The quasiparticle origin of this effect is easily confirmed simply by

reversing the magnetic field, which causes a reversal of the transverse thermal

current. Using this technique, Krishana

et al.

were able to estimate the in-plane

quasiparticle component of the thermal conductivity, which turned out to be about

508

Chapter 10: Thermal Properties

Fig. 10.10.

~ "~" " '' | '

,~~ o

m D

9 D

0.9

0 0

9 ',/ ' 9 -"w --

0 0

....

......

. . .....

0 2 4 6

(Tesla)

Field dependence of the thermal conductivity for an YBa2Cu307 single crystal at 41K. O,

Magnetic field oriented parallel to the

a-b

plane. [:], Magnetic field oriented perpendicular to

the

a-b

plane and increasing in magnitude. 9 Field oriented perpendicular to the

a-b

plane

and decreasing in magnitude. [After Peacor

et al.

(1991b).]

Fig. 10.11.

---16

(b) Untwinned

A T=61K

T=31K

I I .~ I

2 4

H(Tesla)

Field dependence of the thermal conductivity of an untwinned crystal of YBa2Cu307_6. Solid

lines are fits to Eq. 34. The dashed line is an unsuccessful fit assuming the functional

dependence of Eq. 33. [After Richardson

et al.

(1991).]

C. Thermal Conductivity 509

Fig. 10.12.

•

Experimental arrangement for the Righi-Leduc effect used to determine asymmetric scattering

of quasiparticles in a crystal of YBazCu307_ ~.

50% of the total thermal conductivity below T c. The other 50% of heat current is

then due to phonons.

References for Section C

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19, 1228 (1964).

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H. Bougrine, S. Sergeenkov, and M. Ausloos,

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Hermann, Phys. Rev. B

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Phys. Kondens. Mater.

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Phys. Rev. B

38, 2892 (1988).

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Phys. Rev. B

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Progress in High

Temperature Superconductivity

(R. Nikolsky, Ed.), Vol. 25, p. 462. World Scientific, Singapore,

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Sov. Phys. Doklady

3, 116 (1958).

U. Gottwick, R. Held, G. Sparn, E Steglich, H. Rietschel, D. Ewert, B. Renker, W. Bauhoffr, S. von

Molnar, M. Wilhelm, and H. E. Hoenig,

Europhys. Lett.

4, 1183 (1987).

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Phys. Rev. B

40, 9389 (1989).

A. V. Inyushkin, A. N. Taldenkov, S. Yu. Shabanov, L. N. Demyanets, and T. G. Uvarova,

J. Supercond.

7, 331 (1994).

510 Chapter 10: Thermal Properties

A. Jezowski and J. Klamut, in Studies of High-Temperature Superconductors (A. Narlikar, Ed.), Vol. 4,

p. 263. Nova Science Publishers, New York, 1990.

L. P. Kadanoff and P. C. Martin, Phys. Rev. 124, 670 (1961).

P. G. Klemens, in Thermal Conductivity (R. E Tye, Ed.), Vol. 1. Academic Press, London, 1965.

K. Krishana, J. M. Harris, and N. P. Ong, Phys. Rev. Lett. 75, 3529 (1995).

K. Maki, Phys. Rev. 158, 397 (1967).

D. T. Morelli, G. L. Doll, J. Heremans, M. S. Dresselhaus, A. Cassanho, D. R. Gabbe and H. E Jenssen,

Phys. Rev. B 41, 2520 (1990).

T. T. M. Palstra, B. Batlogg, L. E Schneemeyer, and J. V. Waszczak, Phys. Rev. Lett. 64, 3090 (1990).

S. D. Peacor and C. Uher, Phys. Rev. B 39, 11559 (1989).

S. D. Peacor, R. A. Richardson, E Nori, and C. Uher, Phys. Rev. B 44, 9508 (1991 a).

S. D. Peacor, J. L. Cohn, and C. Uher, Phys. Rev. B 43, 8721 (1991 b).

R. A. Richardson, S. D. Peacor, E Nori, and C. Uher, Phys. Rev. Lett. 67, 3856 (1991).

M. Sera, S. Shamoto and M. Sato, Solid State Commun. 74, 951 (1990).

S. Sergeenkov and M. Ausloos, Phys. Rev. B 52, 3614 (1995).

G. Spare, M. Baenitz, S. Horn, E Steglich, W. Assmus, T. Wolf, A. Kapitulnik, and Z. X. Zhao, Physica

C 162-164, 508 (1989).

L. Tewordt, Phys. Rev. 128, 12 (1962).

L. Tewordt and Th. Wolkhausen, Solid State Commun. 70, 839 (1989).

C. Uher, J. Superconductivity 3, 337 (1990).

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159. World Scientific, Singapore, 1992.

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Thermoelectric and Thermomagnetic Effects

Ctirad Uher and Alan B. Kaiser

a. Introduction

Thermoelectric and thermomagnetic phenomena are sensitive probes of the

nature of electronic states and interaction processes within a conductor. As

such they are useful tools to study the band structure and carrier dynamics of

conducting solids. In superconductors, apart from providing valuable insight into

the carrier transport above the superconducting transition temperature To, the

thermoelectric and thermomagnetic effects are important in assessing the

dynamics of vortices and quasiparticles. In this section we illustrate the spectrum

of experiments and the physical parameters obtained from measurements of the

D. Thermoelectric and Thermomagnetic Effects 511

thermoelectric and thermomagnetic effects in superconductors. The terminology

and designation of the effects is analogous to the thermoelectric and thermo-

magnetic effects in the normal metals and semiconductors; see Table 10.3. The

reader may find useful and more in-depth treatment of the thermoelectric and

thermomagnetic effects in superconductors in a monograph by Huebener (1979),

and in the review articles by Kaiser and Uher (1991) and Freimuth (1992).

When discussing thermomagnetic effects at temperatures below T c, one

should keep in mind that the physics refers to the mixed state of a superconductor

characterized by the presence of the flux lines or vortices. It is the dissipative

motion of these vortices that gives rise to the transverse thermomagnetic effects.

The vortex itself, to a first approximation, is viewed as a rigid tube of normal

phase of radius ~ (coherence length) threading the superconducting phase. The

core of the vortex is screened from the superconducting surrounding by the

encircling screening supercurrent Jsc of magnitude

Jsc - (~bo/2rC#o23)K1(r/2),

(35)

where K 1 (r/2) is the first-order modified Bessel function, 4)o is the flux quantum,

2 stands for the penetration depth, and r is the distance from the center of the

vortex. The functional form of Jsc together with its asymptotic behaviors for

r << 2 and r >> 2 are shown in Fig. 10.13. Although this description is quite

adequate for the conventional superconductors, the vortices in the high-tempera-

ture superconductors (HTS) are much less rigid and may attain a partly two-

Table 10.3.

Galvanomagnetic and thermomagnetic effects.

Effect

Electric Electric Temp. Heat Magnetic

Symbol field (E) current (I) gradient (VT) current (q) field (B)

Resistivity p Meas. x Appl. x 0

Magnetores.

longitudinal p(Bii ) Meas. x Appl. x 0

Magnetores.

transverse

p(B•

Meas. x Appl. x 0

Thermal

conductivity ~c 0 Meas. x

Hall effect R H Meas. y Appl. x 0

Seebeck S Meas. x 0 Appl. x

Magneto-

Seebeck

S(B)

Meas. x 0 Appl. x

Peltier H Appl. x 0

Ettingshausen e Appl. x Meas. y

Righi-Leduc R L Meas. y

Nemst Q Meas. y 0 Appl. x

Appl. x

Meas. x

Appl. x

Appl. z

Appl. x, y, z

Appl. z

512

Chapter

10:

Thermal Properties

Fig. 10.13.

10.0

8.0

6.0

dse

4.0

2.0 J- e -'la

o.o

Radial dependence of the screening current Jsc. Asymptotic behavior for r << 2 and for r >> 2

is indicated. The actual numerical value of Jsc is obtained by multiplying the vertical axis by

~bo/(2n#o)? )

2.62

x 10-1~ 3

Am, where 2 is the penetration depth.

dimensional character, especially in the more anisotropic cuprates. Nonetheless,

in describing the thermomagnetic effects, we adhere to the picture of vortices as

rigid tubes. We first introduce the Lorentz force and the thermal force as the two

driving forces responsible for the motion of vortices.

b. Lorentz and Thermal Forces

A flow of transport current density J exerts a Lorentz force F L on a unit length of

the vortex line given by

FL = J • +o, (36)

where ~0 is the flux quantum. In the configuration of Fig. 10.14 with the

transport current density

along the x-axis and the vortex parallel to the z-axis,

the Lorentz force points down along the negative y-axis. Assuming no pinning,

vortices move in response to the Lorentz force but their motion is damped by the

viscous drag force

= --~/)L,y,

where r/is the viscous drag coefficient and

VL,y

the steady-state vortex velocity. The motion of the free vortices,

called flux flow

viscous flow

of vortices, is thus governed by the equation

F L + fn = 0. (37)

Thermoelectric and Thermomagnetic Effects SI3

Fig. 10.14.

I ~ m im u

The Lorentz force F e acting on a vortex oriented parallel to the z-axis arising from the presence

of the transport current density along the x-axis.

According to Josephson, vortices moving with a velocity v L generate the

macroscopic electric field

E = B x v L, (38)

where B is the magnetic induction, B--nq~ 0 with n the density of vortices.

Combining Eqs. (36)-(38), the electric field in the x-direction becomes

E x = (c~oBz/~l)Jx.

(39)

Equation (39) leads to the electrical resistivity

called flux flow resistivity

pf, given

pf = Ex/Jx = dPoBz/r l.

(40)

At the magnetic field equal to the upper critical field, Bc2 , the material becomes

normal,

Pn -- ~b0Bc2/r/.

(41)

Eliminating ~/between Eqs. (40) and (41) yields

pf - PnBz/Bc2 cx Pn ~2/a20"

(42)

The approximation on the RHS of Eq. (42) follows from a well-known relation

~-(dPo/2gBc2) 1/2,

and the expression for the vortex lattice parameter

a 0 -(~o/B) 1/2.

Equation (42) basically states that the flux flow resistivity is

proportional to the fraction of the normal volume occupied by the vortex cores in

a unit cell of the vortex lattice.

514

Chapter 10: Thermal Properties

If, instead of the external electric current, the vortex is subjected to a

thermal gradient, then there is a

thermal force,

Fth, acting on the vortex. In

reference to Fig. 10.15, the thermal force acts along the x-axis and is given by

Fth = -SoVxT (43)

where S o is the transport entropy per unit length of vortex. The origin of the

thermal force is in the temperature dependence of the penetration length, which is

larger at the hot end of the sample than at the cold end. This causes stronger

repulsion of vortices near the hot end, which gives rise to their motion down the

thermal gradient. The two forces, the Lorentz force and the thermal force, are at

the heart of the thermomagnetic phenomena in the mixed-state of superconduc-

tors. We start with the so-called longitudinal effects, the Seebeck effect and the

Peltier effect.

c. Seebeck Effect

A metal with no electric current flowing, and subjected to a longitudinal

temperature gradient Vx T in the x-direction, can develop a longitudinal electric

field proportional to the temperature gradient

E x = SVx T (44)

a phenomenon called the Seebeck effect. The proportionality constant S is called

the Seebeck coefficient or, equivalently, the thermopower. This coefficient should

not be confused with the transport entropy S 0 of Eq. (43). Assuming the sample

Fig. 1035.

/// I r

, -~|

....

The thermal force Fth on a vortex oriented parallel to the z-axis arising from the presence of a

temperature gradient VxT.

D. Thermoelectric and Thermomagnetic Effects

515

geometry of a homogeneous bar with a temperature difference AT between its

ends, the corresponding Seebeck voltage across the length of the sample is

AV = S AT. (45)

The sign of the thermopower corresponds to the sign of the dominant charge

carrier, positive for holes and negative for electrons.

1. Normal State

In metals, semiconductors, and in superconductors above T c, the Seebeck effect

arises as a consequence of the asymmetry in the diffusion of the electrons and

holes along the temperature gradient. In metals, within the free-electron approx-

imation, the thermopower becomes

S = (rcz/2)(kB/e)(T/TF) = 142(T/TF)

(/~V/K). (46)

Note a T-linear dependence and a very small magnitude of S due to a typical

Fermi temperature T v >_ 104 K. In semiconductors, a corresponding expression

for the diffusion thermopower is

S = (kB/e)(ec/k B T) = 86(ec/k B T)

(/~V/K). (47)

Since the conduction band energy e c >> k B T, S is large and varies as T -1 . Exact

quantitative evaluation of the thermopower is often difficult because other

contributions such as the phonon-drag effect, more than one type of scatterer,

and strong dependence on the particular electronic structure complicate the

physical picture. In disordered solids where elastic disorder scattering dominates,

the thermopower is approximately linear except for a characteristic change of

slope near 50 K due to mass enhancement by a factor (1 + 2), where 2 stands for

the dimensionless electron-phonon coupling constant.

Conventional superconductors are metals; hence, their thermopower at

T > T c reflects the characteristic metallic features discussed earlier. Furthermore,

because the electron-phonon interaction is the driving mechanism of super-

conductivity, large enhancement in the thermopower at low temperatures is

expected. An example of such an enhancement in a Chevrel-phase superconduc-

tor is shown in Fig. 10.16.

In addition to their large structural anisotropy, high-temperature cuprates

possess lower carrier density than typical metals and are very sensitive to doping

and structural defects. Hence, one would expect the normal-state thermopower to

be somewhat larger than the typical metallic value and perhaps more complex. In

reality, the thermopower in HTS presents a surprisingly clear pattern, and this

allows one to generalize the behavior across the spectrum of cuprates rather than

to deal with idiosyncrasies of each perovskite family separately. The reader

wishing to survey existing thermopower studies is referred to review articles

dealing with the subject, such as that by Kaiser and Uher (1991).